Probabilistic Modelling is Sufficient for Causal Inference

Abstract: Causal inference is a key research area in machine learning, yet confusion reigns over the tools needed to tackle it. There are prevalent claims in the machine learning literature that you need a bespoke causal framework or notation to answer causal questions. In this paper, we want to make it clear that you \emph{can} answer any causal inference question within the realm of probabilistic modelling and inference, without causal-specific tools or notation. Through concrete examples, we demonstrate how causal questions can be tackled by writing down the probability of everything. Lastly, we reinterpret causal tools as emerging from standard probabilistic modelling and inference, elucidating their necessity and utility.

• The paper shows that augmenting probabilistic models with intervened and counterfactual variables enables standard inference for causal queries.
• It provides formal constructions for interventional, counterfactual, and adjustment queries, unifying Bayesian network methods with do-calculus results.
• The results imply that practical causal reasoning can rely solely on careful probabilistic modelling, simplifying both theoretical and computational approaches.

Probabilistic Modelling as a Sufficient Framework for Causal Inference

Overview

The paper "Probabilistic Modelling is Sufficient for Causal Inference" (2512.23408) systematically argues that probabilistic modelling, in the sense of standard Bayesian Networks and joint density models, provides a complete and sufficient framework for causal inference tasks, including intervention and counterfactual reasoning. This position challenges the prevailing narrative that standard tools from probability and statistics are inherently insufficient for causal inference and must be augmented with separate causal frameworks such as Structural Causal Models (SCMs) or the do-calculus. The authors contest this dichotomy by demonstrating, both formally and constructively, that standard probabilistic models—if constructed to explicitly incorporate both observed and hypothetical (intervened or counterfactual) variables—can answer all standard causal queries without additional formal machinery.

Central Thesis and Argument

The central claim is that the apparent insufficiency of probabilistic methods for causal inference arises not from any intrinsic limitation of probability theory, but from an overly narrow application of probabilistic models that focus only on observational distributions. When one considers joint probabilistic models over all relevant random variables—including those corresponding to intervened-upon or counterfactual worlds—the tools of statistical inference and conditioning can express and compute interventional and counterfactual distributions as required for causal inference.

The formal recipe is to augment the probabilistic model to simultaneously describe the variables for both the observed world and for all hypothetical intervention or counterfactual settings of interest. Under this "twin model" perspective, all causal queries become standard probabilistic queries (e.g., conditional expectations or marginals) in an enlarged probability space. The authors provide detailed constructions and algebraic derivations for the process of expressing interventional and counterfactual distributions using standard Bayesian network operations, including interventions and twin networks, without the need for specialized calculus or extended notational frameworks.

Formal Results and Technical Contributions

The paper contains precise constructions demonstrating the sufficiency of probabilistic models for three central classes of causal queries:

• Interventional queries: Marginals or conditionals on variables under hypothetical interventions (the "do-operator") can be obtained by constructing an augmented model where the structural equations or conditional distributions for intervened variables are manually replaced to reflect the intervention, and then performing probabilistic inference on this model.
• Counterfactual queries: Counterfactual estimands (e.g., "what would $Y$ have been, had $X$ been set to $x'$, in an individual with observed values $X=x, Y=y$?") are constructed using joint models over both observed and counterfactual variables, with appropriate coupling of latent variables or sources of randomness. Inference then reduces to conditioning and marginalization in this twin model.
• Identifiability and adjustment formulas: The paper provides a formal proof that standard adjustment formulas (such as parent adjustment or backdoor adjustment) can be derived directly within the probabilistic framework, consistent with the rules of the do-calculus.

These derivations are exemplified through detailed worked examples involving log-normal Bayesian networks modelling medical intervention scenarios (e.g., aspirin for headache duration). The construction is extended to cover arbitrary probabilistic interventions and "soft" interventions, demonstrating broad applicability.

Critical Evaluation of Competing Claims

A significant portion of the paper analyzes the literature on causal inference, identifying numerous instances where claims are made regarding the insufficiency of probabilistic/statistical or "machine learning" methods for causal tasks. The authors argue that such claims are primarily linguistic or notational: often, "statistical" methods are implicitly defined as those which only fit the observed data distribution, without reference to model-based reasoning about interventions or counterfactuals. However, standard probabilistic tools are not so restricted, and constructing probabilistic models over augmented spaces suffices to recover the full range of causal queries designated in the conventional hierarchy (association, intervention, counterfactual).

This critique is supported by an exhaustive appendix collecting and contextualizing quotations from the literature. The authors highlight that, when made concrete, causal inference is ultimately an inferential problem in a suitably specified probabilistic or Bayesian model.

Implications

Practical Implications

The practical implication is a significant conceptual simplification for practitioners: standard probabilistic programming, Bayesian inference machinery, and joint generative models suffice for causal inference as long as they are used to model all relevant random variables (including interventional/counterfactual ones) and the necessary independence/coupling relations. This explicitly shifts the focus from inventing new formal frameworks to careful, explicit model specification at the probabilistic level. Probabilistic programming languages can, in principle, capture all aspects of interventional, counterfactual, and causal inference without recourse to new calculus or formal languages.

Theoretical Implications

On the theoretical side, the work unifies causal inference with statistical inference, eliminating the need for a conceptual gulf between model-based statistics and graphical/structural causal frameworks. This reframing also emphasizes that all identifiability results, adjustment formulas, and rules of do-calculus can be rendered as consequences of standard probability plus explicit model augmentation.

The approach also clarifies which aspects are truly causal: the substantive content is in model specification (i.e., which mechanisms are modified, which variables are coupled, how interventional and observed variables relate), not in the notation or the requirement for non-standard inference procedures.

Future Directions

Looking forward, this paradigm invites a focus on how to specify, fit, and reason about increasingly complex joint probabilistic models, including those with high-dimensional and structured random variables (e.g., images, sequences, computational graphs), latent variables, and hierarchical dependency. The prospect for unified causal and statistical inference in modern machine learning, including deep generative models and probabilistic programming, is thereby endorsed. Additionally, the sufficiency program established opens questions about model classes, identifiability, and computational tractability: while probabilistic modelling is sufficient, practical inference in high-dimensional, partially discrete, and latent-variable models may nonetheless require specialized computational strategies.

Conclusion

"Probabilistic Modelling is Sufficient for Causal Inference" (2512.23408) provides a thorough formal and conceptual demonstration that joint probabilistic models—extending the standard space to include intervened and counterfactual variables—are a complete framework for all core tasks in causal inference. The paper thus refutes the claim that additional formal tools beyond probability and Bayesian reasoning are required. Instead, it shifts attention to careful model augmentation and specification, providing theoretically and practically significant simplification and unification for both the understanding and application of causal inference in artificial intelligence and statistics.